Suppose that $N$ pairs of socks are put in a washing machine, with each sock having one mate. If the washing machine randomly eats socks, and at the end of the wash returns a random number $K$ of socks where $0 \leq K \leq 2N$, where each $K$ is equally probable, what is the expected number of complete pairs of returned socks?
Just from working out the first few values of $N$, I conjecture that the answer is $N/3$, but I am not sure how to prove it for all values of $N$.
Just to be clear, this is the expected value for an infinite number of trials where each $K$ is equally probable, not the expected value for an infinite number of trials where $K$ is fixed, whereupon the answer is $\displaystyle{K \choose 2}/(2N-1)$.
Best Answer
Let $X$ be the number of pairs of socks that get returned.
You already know that $\mathbb{E}[X|K] = \dfrac{1}{2N-1}\dbinom{K}{2}$ and that $K \sim \text{Uniform} \ \overline{0,2N}$
Thus, $\mathbb{E}[X] = \mathbb{E}[\mathbb{E}[X|K]] = \mathbb{E}\left[\dfrac{1}{2N-1}\dbinom{K}{2} \right] = \dfrac{1}{2N-1}\displaystyle\sum_{k = 0}^{2N}\dfrac{1}{2N+1}\dbinom{k}{2}$ $= \dfrac{1}{(2N-1)(2N+1)}\dbinom{2N+1}{3} = \dfrac{1}{(2N-1)(2N+1)}\dfrac{(2N+1)(2N)(2N-1)}{6} = \dfrac{N}{3}$.